Question

Найдите общий вид первообразных для функций:f(x)=9-sinx f(x)=6x^5-x^10
f(x)=12/x^7
f(x)=8^x
f(x)=cos(5x)
f(x)=20/√4x-3
f(x)=1/(6x-11)^3

Answer 1

$1)f(x) = 9 - \sin(x) \\ F(x) = \int\limits(9 - \sin(x) )dx = \\ = 9x + \cos(x) + c$

$2)f(x) = 6 {x}^{5} - {x}^{10}$

$F(x) = \int\limits(6 {x}^{5} - {x}^{10} )dx = \\ = \frac{6 {x}^{6} }{6} - \frac{ {x}^{11} }{11} + c = {x}^{6} - \frac{ {x}^{11} }{11} + c$

$3)f(x) = \frac{12}{ {x}^{7} }$

$F(x) = \int\limits12 {x}^{ - 7} dx = \\ = 12 \times \frac{ {x}^{ - 6} }{( - 6)} + c = - \frac{2}{ {x}^{6} } + c$

$4)f(x) = {8}^{x}$

$F(x) = \int\limits {8}^{x} dx = \frac{ {8}^{x} }{ ln(8) } + c$

$5)f(x) = \cos(5x)$

$F(x) = \int\limits \cos(5x) dx = \\ = \frac{1}{5} \int\limits \cos(5x) d(2x) = \frac{1}{5} \sin(5x) + c$

$6)f(x ) = \frac{20}{ \sqrt{4x - 3} }$

$F(x) = \int\limits \frac{20}{ \sqrt{4x - 5} } dx = \int\limits \frac{5 \times 4}{ \sqrt{4x - 5} } = \\ = \int\limits \frac{5d(4x - 5)}{ {(4x - 5)}^{ \frac{1}{2} } } =5 \times \frac{ {(4x - 5)}^{ \frac{1}{2} } }{ \frac{1}{2} } = \\ = 2 \sqrt{4x - 5} + c$

$7)f(x) = \frac{1}{ {(6x - 11)}^{3} }$

$F(x) = \int\limits \frac{dx}{ {(6x - 11)}^{3} } = \frac{1}{6} \int\limits {(6x - 11)}^{ - 3} d(6x - 3) = \\ = \frac{1}{6} \times \frac{ {(6x - 11)}^{ - 2} }{( - 2)} + c = - \frac{1}{12 {(6x - 11)}^{2} } + c$